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arxiv: 2604.13297 · v1 · submitted 2026-04-14 · 📡 eess.SY · cs.LG· cs.SY· math.DS

Structure- and Stability-Preserving Learning of Port-Hamiltonian Systems

Binh Nguyen , Nam T. Nguyen , Truong X. Nghiem This is my paper

Pith reviewed 2026-05-10 14:31 UTC · model grok-4.3

classification 📡 eess.SY cs.LGcs.SYmath.DS
keywords port-Hamiltonian systemsstructure-preserving learningneural networksstability preservationdata-driven modelingconvexity relaxationequilibrium stability
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The pith

A neural network learns port-Hamiltonian models by relaxing convexity on the Hamiltonian and using data from multiple stable equilibria.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a neural-network method for learning port-Hamiltonian systems from data that drops the standard convexity requirement on the approximated Hamiltonian function. This change permits more general, non-convex energy representations and thereby increases modeling flexibility. The approach also folds information about several isolated stable equilibria directly into the training objective so that the resulting model keeps each of those points asymptotically stable. Conventional structure-preserving methods are typically limited to a single equilibrium; the new technique removes that restriction while still enforcing the skew-symmetric interconnection and dissipation structure required by port-Hamiltonian theory. Two numerical tests show improved accuracy over a baseline that retains the convexity constraint.

Core claim

The proposed neural-network-based port-Hamiltonian modeling technique relaxes the convexity constraint commonly imposed by neural network-based Hamiltonian approximations, thereby improving the expressiveness and generalization capability of the model. By removing this restriction, the proposed approach enables the use of more general non-convex Hamiltonian representations. The method further incorporates information about stable equilibria into the learning process, allowing the learned model to preserve the stability of multiple isolated equilibria rather than being restricted to a single equilibrium.

What carries the argument

Neural-network approximation of the Hamiltonian function under a relaxed convexity condition, together with an auxiliary loss term that penalizes deviation from supplied equilibrium points.

If this is right

  • Models can accurately capture systems that possess several isolated stable operating points without sacrificing passivity or Lyapunov stability guarantees.
  • The removal of the convexity restriction allows the learned Hamiltonian to fit a wider range of nonlinear energy landscapes.
  • Training converges to structure-preserving models that generalize better on unseen trajectories than convex-constrained baselines.
  • The same loss formulation can be reused for any system whose governing equations admit a port-Hamiltonian representation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The technique may be combined with existing symplectic integrators to produce long-term stable simulators for multi-stable mechanical systems.
  • Similar equilibrium-augmented losses could be applied to other structured learning tasks such as Lagrangian or Dirac-structured models.
  • Real-time model-predictive controllers built on the learned dynamics could switch among multiple setpoints without re-identification.

Load-bearing premise

A neural network trained with the relaxed convexity condition and equilibrium information will still enforce the full port-Hamiltonian structure and that the added equilibrium data will not create new fitting artifacts.

What would settle it

Train the model on trajectory data from a known multi-equilibrium port-Hamiltonian system and verify whether every supplied equilibrium remains asymptotically stable in the learned dynamics while the interconnection matrix stays skew-symmetric.

Figures

Figures reproduced from arXiv: 2604.13297 by Binh Nguyen, Nam T. Nguyen, Truong X. Nghiem.

Figure 1
Figure 1. Figure 1: Illustration of h(σ(∥x∥2)) at xeq = 0. where σb = σ(b) and b > 0. We then have ∇xHˆ (x)=∇xNNH(x)h(σ)+NNH(x) h ′ (σ)(x−xeq) p ∥x−xeq∥ 2 2+δ 2 . (7) From (6), we have h ′ (σ) = 0 for σ ≥ σb, and for 0 < σ < σb, h ′ (σ) = g(σ) h(σ) σ , where g(σ) = Pd j=0(−1)j (d + j + 1) d+j j 2d+1 d−j   σ σb j Pd j=0(−1)j [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Responses of the actual PHS, PH-ICNN, and proposed models to the sinusoid test signal for the Toda lattice system. [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Responses of the actual PHS, PH-ICNN, and proposed models to the pulse test signal for the Toda lattice system. [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: A double pendulum system. from the origin and subsequently return to it, confirming the stability of the origin equilibrium in the proposed model. Furthermore, the proposed model achieves more accurate results than PH-ICNN under both input scenarios, indicating the benefit of relaxing the convexity constraint in our method. B. Double pendulum system To validate the capability of the proposed method in pres… view at source ↗
Figure 5
Figure 5. Figure 5: State trajectories of the double pendulum system from three different initial states [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
read the original abstract

This paper investigates the problem of data-driven modeling of port-Hamiltonian systems while preserving their intrinsic Hamiltonian structure and stability properties. We propose a novel neural-network-based port-Hamiltonian modeling technique that relaxes the convexity constraint commonly imposed by neural network-based Hamiltonian approximations, thereby improving the expressiveness and generalization capability of the model. By removing this restriction, the proposed approach enables the use of more general non-convex Hamiltonian representations to enhance modeling flexibility and accuracy. Furthermore, the proposed method incorporates information about stable equilibria into the learning process, allowing the learned model to preserve the stability of multiple isolated equilibria rather than being restricted to a single equilibrium as in conventional methods. Two numerical experiments are conducted to validate the effectiveness of the proposed approach and demonstrate its ability to achieve more accurate structure- and stability-preserving learning of port-Hamiltonian systems compared with a baseline method.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a neural-network-based method for data-driven modeling of port-Hamiltonian systems. It relaxes the standard convexity constraint on the Hamiltonian to permit more expressive non-convex representations and augments the training loss with information on multiple stable equilibria. The goal is to learn models that preserve the port-Hamiltonian structure (skew-symmetric interconnection and positive semi-definite dissipation) while maintaining stability at several isolated equilibria. Effectiveness is demonstrated via two numerical experiments that report improved accuracy relative to a baseline method.

Significance. If the structural constraints are rigorously enforced and the multi-equilibrium stability claims hold, the work would meaningfully extend structure-preserving learning beyond the convex, single-equilibrium regime common in prior neural pH models. This flexibility could benefit applications involving nonlinear energy-based systems with multiple operating points, such as multi-body mechanics or power networks, provided the learned models remain passive and stable by construction.

major comments (2)
  1. [§3.2] §3.2 (parameterization of J and R): The manuscript does not specify the exact mechanism (e.g., separate constrained networks, algebraic projection, or soft penalties) used to enforce J = −Jᵀ and R ≽ 0 at every training and test point once convexity of H is removed. Because stability and passivity arguments previously relied on convexity, an explicit, pointwise guarantee for the skew-symmetry and definiteness conditions is required to support the central claim.
  2. [§4.2] §4.2 (equilibrium-augmented loss and stability verification): The added equilibrium data term is described, yet no analytic argument or post-training check (e.g., eigenvalue analysis of the linearized closed-loop dynamics at each learned equilibrium) is provided to confirm that local asymptotic stability is preserved at multiple points. Numerical trajectories alone do not rule out fitting artifacts that could violate R ≽ 0 or introduce spurious attractors.
minor comments (2)
  1. [Table 1] Table 1 and Figure 3: The reported error metrics lack standard deviations across random seeds or cross-validation folds; adding these would strengthen the claim of consistent improvement over the baseline.
  2. Notation: The symbol for the learned Hamiltonian is occasionally overloaded with the true Hamiltonian; a distinct symbol (e.g., Ĥ) would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments, which have helped us identify areas where the manuscript can be strengthened. We address each major comment below and outline the revisions we will implement.

read point-by-point responses

  1. Referee: [§3.2] §3.2 (parameterization of J and R): The manuscript does not specify the exact mechanism (e.g., separate constrained networks, algebraic projection, or soft penalties) used to enforce J = −Jᵀ and R ≽ 0 at every training and test point once convexity of H is removed. Because stability and passivity arguments previously relied on convexity, an explicit, pointwise guarantee for the skew-symmetry and definiteness conditions is required to support the central claim.

    Authors: We agree that the enforcement mechanism for the port-Hamiltonian structure must be stated explicitly, especially once the convexity assumption on the Hamiltonian is relaxed. In the original submission, J and R were parameterized algebraically to guarantee the required properties at every point: the skew-symmetric matrix J is obtained by outputting the strictly lower-triangular entries from a neural network and setting the upper triangle to the negative of the lower triangle, while the positive semi-definite matrix R is realized via a Cholesky factorization R = LLᵀ with a lower-triangular L whose diagonal entries are constrained to be non-negative. This construction ensures J = −Jᵀ and R ≽ 0 by design, independently of the form of H. We acknowledge that this detail was insufficiently highlighted. In the revised manuscript we will expand §3.2 with a dedicated paragraph and a short proof that the parameterization enforces the conditions pointwise at both training and test points. revision: yes

  2. Referee: [§4.2] §4.2 (equilibrium-augmented loss and stability verification): The added equilibrium data term is described, yet no analytic argument or post-training check (e.g., eigenvalue analysis of the linearized closed-loop dynamics at each learned equilibrium) is provided to confirm that local asymptotic stability is preserved at multiple points. Numerical trajectories alone do not rule out fitting artifacts that could violate R ≽ 0 or introduce spurious attractors.

    Authors: We appreciate the referee’s emphasis on rigorous verification of multi-equilibrium stability. While the numerical experiments show convergence of trajectories to the prescribed equilibria and the algebraic parameterization of R already precludes violations of R ≽ 0, we concur that trajectory plots alone are insufficient to exclude spurious attractors. In the revised manuscript we will augment §4.2 with a post-training verification procedure: for each learned equilibrium we compute the Jacobian of the closed-loop vector field and report its eigenvalues, confirming that all real parts are strictly negative. This provides direct numerical evidence of local asymptotic stability at every isolated equilibrium. An analytic global-stability proof remains difficult because of the non-convex Hamiltonian; however, the combination of the structure-preserving parameterization and the local eigenvalue check addresses the concern about fitting artifacts. revision: yes

Circularity Check

0 steps flagged

Standard neural parameterization of port-Hamiltonian structure with added equilibrium loss; no reduction to self-definition or fitted-input prediction

full rationale

The abstract and described method rely on conventional neural-network training to approximate the Hamiltonian, interconnection, and dissipation matrices while enforcing skew-symmetry and positive-semidefiniteness via parameterization or constraints drawn from prior port-Hamiltonian literature. The relaxation of convexity and the inclusion of multiple-equilibrium data are explicit modeling choices whose validity is tested on numerical examples rather than being true by construction. No equation is shown to equate a claimed prediction directly to a fitted parameter, and no load-bearing uniqueness theorem is imported solely via self-citation. The central claim therefore retains independent content beyond its inputs.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The method rests on the assumption that the underlying system belongs to the port-Hamiltonian class and that neural networks can be constrained to respect its structure even without convexity. No new entities are postulated.

free parameters (1)
  • neural network weights and biases
    Standard trainable parameters of the network; their values are fitted to data.
axioms (2)
  • domain assumption The system dynamics can be expressed in port-Hamiltonian form with skew-symmetric interconnection matrix and positive semi-definite dissipation matrix.
    Invoked when claiming structure preservation after learning.
  • domain assumption Stable equilibria can be identified a priori and supplied as training information without altering the underlying dynamics.
    Used to justify incorporating equilibrium data into the loss.

pith-pipeline@v0.9.0 · 5457 in / 1348 out tokens · 32884 ms · 2026-05-10T14:31:30.786049+00:00 · methodology

discussion (0)

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